Fig. 12.16

Diagrammatic representation of the scattering matrix with a bound state pole,

An equation for T can be derived by assuming that Eq. (12.40) holds everywhere, even at the pole. Substituting Eq. (12.48) into Eq. (12.40), multiplying by - P2, and then taking the limit as P2 —<■ M| eliminate all terms not singular at P2 = M%. Dropping the term T(p',P) from both sides gives Eq. (12.47). Note that, strictly speaking, P) is uniquely defined only at the bound state pole, where P2 = Mg. Alternatively, we may say that Eq. (12.47) does not hold except when P2 = Ai|, and hence it is an eigenvalue equation. The relativistic bound state wave function is defined to be ij(p,p)=AfG(p,p)r(p,p),

where M is a normalization constant, to be defined later. [Note that the normalization of T is defined by (12.48).]


The normalization condition for the bound state wave function can be obtained directly from Eq. (12.40) and the assumed form of the A4-matrix, Eq. (12.48). To this end, note that (12.40) can also be written

where, for compactness, we will suppress all arguments of M, G, and V. The equivalent of (12.40) and (12.50) follows from the fact that they generate the same Born series. In general, V will be real but G will be complex because of the singularities associated with the zeros in its denominator. Hence (12.50) may also be written

where the bar represents the adjoint, which includes complex conjugation and any additional operations (such as multiplication by 70 as in the Dirac theory). Writing (12.51) as and substituting this expression for V under the / in (12.40) give the following equation:

Note, for later use, that substituting V obtained from (12.40) into Eq. (12.51) gives a similar equation for m, m = v+ mgm


Only one of these equations is needed now, and it will be used below threshold (in the neighborhood of the bound state pole) where m and g are real. Substituting Eq. (12.48), written in shorthand as m = -r

into (12.53) or (12.54) gives terms with a double pole at = P2, a single pole, and no pole. The double pole terms occur only on the right-hand side (RHS) of the equation and are double poles = (^^i-^^jr j (rGr)r-r J j (rcvGr)r]

The coefficient of the double pole term must be zero at P2 — M%. Dropping the initial factor of T and the final factor T gives j TGT - J J rvGr = J TG |r - J vGr

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